How to Read Math Expressions

Basic

Expression	How to Read	Translate
$a + b$	a plus b	Translate
$a - b$	a minus b	Translate
$a \cdot b$	a times b	Translate
$a / b$	a over b	Translate
$a > b$	a is greater than b	Translate
$a < b$	a is less than b	Translate
$a = b$	a is equal to b	Translate
$a \neq b$	a is not equal to b	Translate
$a \geq b$	a is greater than or equal to b	Translate
$a \leq b$	a is less than or equal to b	Translate
$x^{2}$	x squared	Translate
$x^{3}$	x cubed	Translate
$x^{n}$	x to the power of n	Translate
$\sqrt{x}$	the square root of x	Translate
$\sqrt[n]{x}$	the n-th root of x	Translate
$\log x$	the logarithm of x (base 10)	Translate
$\ln x$	the natural logarithm of x (base e)	Translate
$e^{x}$	e to the power of x	Translate
$\| x \|$	the modulus of x	Translate
$\overset{―}{x}$	x bar (the average of x)	Translate
$a \pm b$	a plus or minus b	Translate
$\frac{a}{b}$	a over b (fraction)	Translate
$a!$	a factorial	Translate

Numbers

Expression	How to Read	Translate
$- 5$	negative five	Translate
$0.25$	zero point two five	Translate
$1.5$	one point five	Translate

Fractions

Expression	How to Read	Translate
$\frac{1}{2}$	one half	Translate
$\frac{3}{4}$	three quarters	Translate
$\frac{1}{7}$	one seventh	Translate

Powers and Roots

Expression	How to Read	Translate
$3^{2}$	three squared	Translate
$5^{4}$	five to the power of four	Translate
$\sqrt{4}$	the square root of four	Translate
$\sqrt[3]{8}$	the cube root of eight	Translate

Angle Operations

Expression	How to Read	Translate
$\sin θ$	sine of theta	Translate
$\cos θ$	cosine of theta	Translate
$\tan θ$	tangent of theta	Translate
$\csc θ$	cosecant of theta	Translate
$\sec θ$	secant of theta	Translate
$\cot θ$	cotangent of theta	Translate
$\sin^{- 1} x$	arcsine of x or inverse sine of x	Translate
$\cos^{- 1} x$	arccosine of x or inverse cosine of x	Translate
$\tan^{- 1} x$	arctangent of x or inverse tangent of x	Translate
$\sin (α \pm β)$	sine of (alpha plus or minus beta)	Translate
$\cos (α \pm β)$	cosine of (alpha plus or minus beta)	Translate
$\tan (α \pm β)$	tangent of (alpha plus or minus beta)	Translate
$θ \pm ϕ$	theta plus or minus phi	Translate
$θ \cdot ϕ$	theta times phi	Translate
$θ / ϕ$	theta over phi	Translate
$\sinh x$	hyperbolic sine of x	Translate
$\cosh x$	hyperbolic cosine of x	Translate
$\tanh x$	hyperbolic tangent of x	Translate

Logic and Sets

Expression	How to Read	Translate
$a := b$	a is defined by(given as) b	Translate
$S_{1} \Rightarrow S_{2}$	S sub one implies S sub two	Translate
$S_{1} \Leftrightarrow S_{2}$	S sub one is equivalent to S sub two	Translate
${a, b, c}$	the set of a, b, and c	Translate
$a \in A$	a is an element of set A	Translate
$\emptyset or {}$	empty set	Translate
$A \subset B$	A is a proper subset of B	Translate
$A \subseteq B$	A is a subset of B	Translate
$A \cup B$	A union B	Translate
$A \cap B$	A intersection B	Translate
$A ∖ B$	A minus B	Translate
$\equiv$	is equivalent to	Translate
$\exists$	there exists	Translate
$∄$	there does not exist	Translate
$\forall$	for all	Translate
$\exists p \in P, p = 2$	there exists a prime number p such that p equals two.	Translate
$\forall x \in R, x + 0 = x$	for all x in the set of real numbers, x plus zero equals x.	Translate
$N$	the set of all natural numbers	Translate
$Z$	the set of all integers	Translate
$Q$	the set of all rational numbers	Translate
$R$	the set of all real numbers	Translate
$C$	the set of all complex numbers	Translate
$P$	the set of all prime numbers	Translate
$A △ B$	symmetric difference of A and B	Translate
$A^{c}$	complement of A	Translate
$\overset{―}{A}$	closure of A	Translate

Intervals

Expression	How to Read	Translate
$[a, b]$	closed interval from a to b	Translate
$(a, b)$	open interval from a to b	Translate
$[a, b)$	the interval from a to b, inclusive of a and exclusive of b.	Translate
$(a, b]$	the interval from a to b, exclusive of a and inclusive of b.	Translate
$(a, \infty)$	the half line starting at a and extending to infinity, exclusive of a.	Translate
$(- \infty, a)$	the half line starting at negative infinity and extending to a, exclusive of a.	Translate

Factorials and Binomial Coefficients

Expression	How to Read	Translate
$n!$	n factorial (the product of all positive integers up to n)	Translate
$(\binom{n}{k}) or n C_{k}$	n choose k (the number of ways to choose k elements from a set of n elements)	Translate
$P (n, k)$	permutation of n things taken k at a time	Translate

Sum and Product

Expression	How to Read	Translate
$\sum_{n = 1}^{\infty} a_{n}$	the sum of a sub n from n equals 1 to infinity	Translate
$\sum_{k = 0}^{n} (\binom{n}{k})$	the sum from k equals zero to n of n choose k	Translate
$(\sum_{k = 1}^{n} a_{k} b_{k})^{2}$	the square of the sum of the products of a sub k and b sub k from k equals 1 to n	Translate
$\prod_{k = 1}^{n} k$	the product from k equals one to n of k (n factorial)	Translate
$⋃_{i = 1}^{n} A_{i}$	union of sets A sub i from i equals 1 to n	Translate
$⋂_{i = 1}^{n} A_{i}$	intersection of sets A sub i from i equals 1 to n	Translate

Functions

Expression	How to Read	Translate
$φ$	phi (variant)	Translate
$φ (0) = 0$	phi of zero equals zero	Translate
$f : A \to B$	f is a function that maps from the set A to the set B	Translate
$x \mapsto f (x)$	x is mapped to f(x)	Translate
$g : R \to C, θ \mapsto g (θ) := e^{i θ}$	g from R(real numbers) to C(complex numbers), theta maps to g of theta, defined as(given by) e to the i theta	Translate
$f : R \to R, x \mapsto f (x) := 1 + x^{2}$	f from R(real numbers) to R, x maps to f of x, defined as(given by) one plus x squared	Translate
$h : C \to [0, \infty), z \mapsto h (z) := \| z \|$	function h maps complex numbers C to the set of non-negative real numbers from zero to infinity, z maps to function h of z equals the modulus of z	Translate
$im (f)$	image or range of f	Translate
$im (f) = B$	the image(range) of f equals B	Translate
$f : A \to im (f)$	f from A to the image of f	Translate
$f^{- 1}$	inverse function of f	Translate
$f^{- 1} (b) := a$	f inverse of b is defined as(given by) a	Translate
$f_{i, j}$	f sub i comma j	Translate
$f \circ g$	f composed with g	Translate
$\ker (f)$	kernel of f	Translate
$f (x)$	f of x	Translate
$lim_{x \to 0}$	x approaches 0	Translate

Calculus Operation Notations

Expression	How to Read	Translate
$\frac{d}{d x}$	d by dx (the derivation with respect to x)	Translate
$\frac{d}{d x} f (x)$	the derivative of f with respect to x	Translate
$f^{'} (x)$	f prime of x (the derivative of f with respect to x)	Translate
$\int f (x) d x$	the integral of f with respect to x	Translate
$\int_{a}^{b} f (x) d x$	the definite integral of f from a to b with respect to x	Translate
$lim_{x \to a} f (x)$	the limit of f as x approaches a	Translate
$\oint f (x) d x$	the line integral of f with respect to x	Translate

Vector calculus and physics

Expression	How to Read	Translate
$\nabla$	nabla or del	Translate
$\nabla f$	gradient of(nabla) f	Translate
$\nabla \cdot F$	divergence of(nabla dot) F	Translate
$\nabla \times F$	curl of(nabla cross) F	Translate
$\partial$	partial	Translate
$\frac{\partial}{\partial x} f (x)$	the partial derivative of f with respect to x	Translate
$\frac{\partial f}{\partial x}$	partial of f with respect to x	Translate
$\frac{\partial^{2} f}{\partial x^{2}}$	the second partial derivative of f with respect to x squared	Translate
$\frac{\partial}{\partial x_{j}} (\frac{\partial f}{\partial x_{i}})$	The partial derivative with respect to x sub j of the partial derivative of f with respect to x sub i	Translate
$\frac{\partial^{2} f}{\partial x_{j} \partial x_{i}}$	The second partial derivative of f with respect to x sub j and x sub i	Translate
$a \cdot b$	dot product of vectors a and b	Translate
$a \times b$	cross product of vectors a and b	Translate

Matrix Operations

Expression	How to Read	Translate	Pronounce
$A^{T}$	transpose of matrix A	Translate
$A^{- 1}$	inverse of matrix A	Translate

Complex Numbers

Expression	How to Read	Translate
$z = a + b i$	z equals a plus b i (complex number)	Translate
$\| z \|$	modulus of z	Translate
$\arg (z)$	argument of z	Translate
$\overset{―}{z}$	complex conjugate of z	Translate

Differential Equations

Expression	How to Read	Translate	Pronounce
$\frac{d y}{d x} = f (x, y)$	dy by dx equals f of x and y (first-order differential equation)	Translate
$\frac{d^{2} y}{d x^{2}} + p (x) \frac{d y}{d x} + q (x) y = r (x)$	d squared y by dx squared plus p of x times dy by dx plus q of x times y equals r of x (second-order linear differential equation)	Translate

How to Read Math Expressions

Learn to Pronounce Math Symbols and Formulas

Basic

Numbers

Fractions

Powers and Roots

Angle Operations

Logic and Sets

Intervals

Factorials and Binomial Coefficients

Sum and Product

Functions

Calculus Operation Notations

Vector calculus and physics

Matrix Operations

Complex Numbers

Differential Equations

Source

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